Special case: grouplike elements in coalgebras

Relation to differential graded algebras

For corings with a (sometimes semi-)grouplike element one can define many useful notions which do not exist for general corings.

For example, given a semi-grouplike element gg, the tensor algebraΩC=⊕iΩiC\Omega C = \oplus_i \Omega^i C of the coring CC, where ΩiC=C⊗A…⊗AC\Omega^i C = C\otimes_A \ldots \otimes_A C (ii times) over AA can be equipped with a differentialdd of degree +1+1 in a canonical way making it a differential graded algebra:

A special case of this construction is when g=1⊗R1g = 1\otimes_R 1 and CC is the Sweedler coring for a kk-algebra extension R→SR\to S. The dga obtained is the classical Amitsur complex Ω(S/R)\Omega(S/R) for that extension; for this reason the complex ΩC=Ω(C,g)\Omega C = \Omega(C,g) above for any coring CC and semi-grouplike gg is sometimes said to be an Amitsur complex.